El problema de Pillai con números de Padovan y potencias de dos

• García Lomeli, Ana María ^[1] ; Hernández Hernández, Santos ^[1]
  1. [1] Universidad Autónoma de Zacatecas

     Universidad Autónoma de Zacatecas

     México

     
• Localización: Revista Colombiana de Matemáticas, ISSN-e 0034-7426, Vol. 53, Nº. 1, 2019, págs. 1-14
• Idioma: español
• DOI: 10.15446/recolma.v53n1.81034
• Títulos paralelos:
  • Pillai's problem with Padovan numbers and powers of two
• Enlaces
  • Texto completo
• Resumen
  • español

    Sea (Pn)n≥0 la sucesión de Padovan dada mediante P0 = 0, P1 = P2 = 1 y la fórmula de recurrencia Pn+3 = Pn+1 + Pn para todo n ≥ 0. En esta nota estudiamos y resolvemos completamente la ecuación diofánticaPn - 2m = Pn1 - 2m1en enteros no negativos (n, m, n1, m1).

  • English

    Let (Pn)n≥0 be the Padovan sequence given by P0 = 0, P1 = P2 = 1 and the recurrence formula Pn+3 = Pn+1 + Pn for all n ≥ 0. In this note westudy and completely solve the Diophantine equationPn - 2m = Pn1 - 2m1in non-negative integers (n, m, n1, m1).

• Referencias bibliográficas
  • H. Davenport A. Baker, The equations 3X2 - 2 = Y 2 and 8X2 - 7 = Z2, Quart. J. Math. Oxford 20 (1969), no. 2, 129-137.
  • J. J. Bravo, C. A. Gómez, and F. Luca, Powers of two as sums of two k-Fibonacci numbers, Miskolc Math. Notes 17 (2016), no. 1, 85-100.
  • J. J. Bravo, F. Luca, and K. Yazán, On Pillai's problem with Tribonacci numbers and Powers of 2, Bull. Korean Math. Soc. 54 (2017), no....
  • Y. Bugeaud, M. Mignotte, and S. Siksek, Classical and modular approaches to exponential diophantine equations I: Fibonacci and Lucas perfect...
  • K. C. Chim, I. Pink, and V. Ziegler, On a variant of Pillai's problem, Int. J. Number Theory 7 (2017), 1711-1727.
  • K. C. Chim, I. Pink, and V. Ziegler, On a variant of Pillai's problem II, J. Number Theory 183 (2018), 269-290.
  • M. Ddamulira, C. A. Gómez, and F. Luca, On a problem of Pillai with k-generalized Fibonacci numbers and powers of 2, Monatsh. Math., 2018,...
  • M. Ddamulira, F. Luca, and M. Rakotomalala, On a problem of Pillai with Fibonacci and powers of 2, Proc. Indian Acad. Sci. (Math. Sci.) 127...
  • A. Dujella and A. Petho, A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford 49 (1998), no. 3, 291-306.
  • S. Hernández Hernández, F. Luca, and L. M. Rivera, On Pillai's problem with the Fibonacci and Pell sequences, Accepted in the Bol. Soc....
  • A. Herschfeld, The equation 2x - 3y = d, Bull. Amer. Math. Soc. 42 (1936), 231-234.
  • E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II, Izv. Math. 64 (2000),...
  • S. S. Pillai, On ax - by = c, J. Indian Math. Soc. 2 (1936), 119-122.
  • S. S. Pillai, On the equation 2x - 3y = 2X + 3Y , Bull. Calcutta Math. Soc. 37 (1945), 15-20.
  • S. Guzmán Sánchez and F. Luca, Linear combinations of factorials and S-units in a binary recurrence sequence, Ann. Math. Québec 38 (2014),...
  • N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, https://oeis.org/.
  • R. J. Stroeker and R. Tijdeman, Diophantine equations, Computational methods in number theory, Math. Centre Tracts (155), Centre for Mathematics...